A323728 a(n) is the smallest number k such that both k-2*n and k+2*n are squares.

2, 5, 10, 8, 26, 13, 50, 20, 18, 29, 122, 25, 170, 53, 34, 32, 290, 45, 362, 41, 58, 125, 530, 52, 50, 173, 90, 65, 842, 61, 962, 80, 130, 293, 74, 72, 1370, 365, 178, 89, 1682, 85, 1850, 137, 106, 533, 2210, 100, 98, 125, 298, 185, 2810, 117, 146, 113, 370

Offset

1,1

Comments

When n is a prime number, a(n) is greater than all the previous terms.

If n = 4*x*y, then a(n) is the smallest integer solution of the form 4*(x^2 + y^2), with rational values x and y.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Wikipedia, Difference of two squares

Formula

a(n^2) = 2 * n^2.

a(p) = p^2 + 1, for p prime.

a(n) = A063655(n)^2 - 2*n.

a(n) = A056737(n)^2 + 2*n.

a(n!) = A061057(n)^2 + 2*n!.

a(n) = A033676(n)^2 + A033677(n)^2. - Robert Israel, Feb 17 2019

a(n) = Min_{d|n} ((n/d)^2 + d^2). - Ridouane Oudra, Mar 17 2024

Example

For n = 3, a(3) = 10, which is the smallest integer k such that k+2*n and k-2*n are both squares: 10+2*3 = 4^2 and 10-2*3 = 2^2.
For n=1..10, the following {a(n)-2*n, a(n)+2*n} pairs of squares are produced: {0, 4}, {1, 9}, {4, 16}, {0, 16}, {16, 36}, {1, 25}, {36, 64}, {4, 36}, {0, 36}, {9, 49}.

Maple

f:= proc(n) local d;
d:= max(select(t -> t^2 <= n, numtheory:-divisors(n)));
d^2 + (n/d)^2
end proc:
map(f, [$1..100]); # Robert Israel, Feb 17 2019

Mathematica

Array[Block[{k = 1}, While[Nand @@ Map[IntegerQ, Sqrt[k + 2 {-#, #}]], k++]; k] &, 57] (* Michael De Vlieger, Feb 17 2019 *)

Prog

(PARI) a(n) = for(k=2*n, oo, if(issquare(k+2*n) && issquare(k-2*n), return(k)));
(PARI) a(n) = my(d=divisors(n)); vecmin(vector(#d, k, 4*((d[k]/2)^2 + (n/d[k]/2)^2)));

Crossrefs

Cf. A000290, A029744, A033676. A033677, A056737, A061057, A063655, A087711.

Sequence in context: A291933 A332198 A154680 * A140469 A001440 A258779

Adjacent sequences: A323725 A323726 A323727 * A323729 A323730 A323731

Keyword

nonn

Author

Daniel Suteu, Jan 25 2019

Status

approved